AMS 5 (Fall, 2009). Solve for x: 0 3 2x = 3 (.2) x Taking the natural log of both sides, we get Applied Calculus I Practice Final Exam Solution Notes Joe Mitchell ln 0 + 2xln 3 = ln 3 + xln.2 x(2ln 3 ln.2) = ln 3 ln 0 x = 2. Determine which function has a larger value as x : ln 3 ln 0 2ln 3 ln.2 = ln(3/0) ln(9/.2) (a). f(x) = 0.0005 x 3 or g(x) = 52,000 2 x f(x) grows like x 3 (we can ignore the constant factor). g(x) grows like 2 x, and exponential function (again, we can ignore the constant factor). Thus, g(x) has a larger value as x. Another way to say this is that f(x) x g(x) = 0, so that g dominates f as x, which we compute using l Hopital s rule (since it has form ): 0.0005 x 3 x 52,000 2 x = x 0.0005 3x 2 52,000 2 x ln 2 = x 0.0005 6x 52,000 2 x ln 2ln 2 = x 0.0005 6 52,000 2 x ln 2ln 2ln 2 = 0. (b). f(x) = log x 7 or g(x) = 6 log 2 x 2 f(x) grows like log x (since log x 7 = 7log x), while g(x) grows like log 2 x (since log 2 x 2 = 2log 2 x), so g(x) dominates f(x) as x. 3. Find the equation of the line that goes through the point (8,3) and is perpendicular to the plot of the equation 2x+ y = 2. We rewrite 2x+ y = 2: 2y 2 = 2x +, so that it is the equation of the line y = x + 3 2. The slope of this line is ; thus, the slope of a line perpendicular to it is =. Thus, the equation of the line, L, we desire is y = x + b. We find b by requiring that (8,3) lies on line L: 3 = 8 + b, so b =. Thus, line L has equation y = x +. 4. The quantity of moisture in a loaf of bread decreases with time as it sits on the table. Suppose that the moisture, M(t), at time t minutes after being placed on the table decreases according to the function M(t) = Qe kt. If 0% of the moisture is gone at the end of hour, (a). What percentage of the original moisture is present after 30 minutes? We are told that M(60) = (.9)M(0), which implies that Qe 60k = (.9)Q Now, we want to find 60k = ln(.9) k = ln(.9) 60 M(30) M(0) 00 = Qe 30k Q ln(.9) 00 = 00e 30 60 = 00e 2 ln(.9) = 00(.9) /2 (b). How long will it take until the moisture is reduced to 60% of its original quantity?
We want to find t so that M(t) = 0.6M(0) = 0.6Q. Thus, we want t so that Qe kt = 0.6Q, i.e., so that e kt = 0.6. Thus, we want t so that kt = ln(0.6), i.e., 5. Evaluate the following expressions: (a). cos(sin ( 5 )) = t = ln(0.6) k = ln(0.6) ln(0.9) 60 = 60 ln(0.6) ln(0.9) Draw a right triangle and let θ be one of the two angles that is not π/2. Then, θ = sin ( 5 ) if the side opposite to θ is of length 5 and the hypotenuse is of length. Then, the side adjacent to θ is of length 2 5 2 = 96. Thus, cos(sin ( 5 )) = cos θ = 96. (b). cos( 32π 32π 3 ) = Note that 3 = 0π + 2π 3 (i.e., it is 0 revolutions, plus an additional angle 2π/3 (20 degrees). Thus, cos( 32π 3 ) = cos(2π/3) = 2. (Draw a picture! The angle refers to a point in the second quadrant.) 6. On February 0, 990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0. feet. Assuming the next high tide is at exactly 2 noon and that the height of the water is given by a sine or cosine curve, find a formula for the water level (in feet) in Boston as a function of time t, measured in hours from midnight. Let H(t) be the height in feet of the tide in Boston t hours after midnight of February 0, 990. Since H(t) oscillates between 0. and 9.9, we know that the amplitude is (9.9-0.)/2=4.9. The baseline of the oscillation occurs at H = 0. + 4.9 = 5; i.e., it is a sin/cos shifted upwards by 5. Since the high value (max) occurs at t = 0 and the first high (max) after that occurs at t = 2 (noon), we know that the period is 2 hours. Draw a picture! We can view H(t) as a cos function that is shifted upwards by 4.9: 7. Find the asymptotes for the following function: H(t) = 5 + 4.9cos( 2π 2 t) y = 3 9x + 5x2 2x 2 (a). Vertical: The denominator, 2x 2, vanishes at x = /2 and at x = /2, so these two values of x define the two vertical asymptotes. (b). Horizontal: We can rewrite (by dividing numerator and denominator by x 2 ) y = 3 x 2 9 x + 5 2 x 2 from which we see the behavior as x go to infinity. As x +, y goes up to 5/2. As x, y goes down to 5/2. Thus, y = 5/2 is a horizontal asymptote. (c). Sketch a plot of y(x) Note that for x = /2 + ǫ (for tiny ǫ > 0), y is positive, heading to +. Note that for x = /2 ǫ (for tiny ǫ > 0), y is negative, heading to. Note that for x = /2 + ǫ (for tiny ǫ > 0), y is negative, heading to. Note that for x = /2 ǫ (for tiny ǫ > 0), y is positive, heading to +. As x +, y goes up to 5/2. As x, y goes down to 5/2. From these facts, we can sketch the plot. Check yourself by typing plot (3-9x+5x^2)/(2x^2-) 2
into wolframalpha.com, and you will see the plot. (You will see that the function has a local minimum for a value of x > /2; now that we have studied derivatives, we know why! Find the critical points.) 8. The displacement (in feet) of a car moving in a straight line is given by s(t) = 4t 2 + t +, where t is measured in seconds. (a). Find the average velocity of the car over the time interval [9,3]. The average velocity over the interval is s(3) s(9) 3 9 = (4 32 + 3 + ) (4 9 2 9 ) 3 9 = 89ft/sec (b). Find the instantaneous velocity of the car when t = 0. The instantaneous velocity at t = 0 is the it as ǫ goes to zero of s(0 + ǫ) s(0) 0 + ǫ 0 which is the derivative of s(t) evaluated at t = 0: s (t) = 8t +, so s (0) = 8 ft/sec. 9. Find the equation of the tangent line to the curve y x = 3 (x 0) at the point (4,5). The curve is y(x) = 3 + x. Since the slope is given by the derivative function, y (x) = (/2)x /2 = 2 x, we know that the slope of the tangent line at (4,5) is y (4) = /4. Thus, the equation of the tangent line at (4,5) has the form y = (/4)x+b, and we can find b using the fact that the line must pass through the point (4,5): 5 = (/4) 4+b, implying that b = 4. Thus, the equation of the tangent line at (4,5) is y = (/4)x + 4. 0. If h(b) = x 2 b 3 + 3b 2 x2 b 3, find h (). Also find h (x). h (b) = x 2 3b 2 + 6b x 2 ( 3)b 4 = 3x 2 b 2 + 6b + 3x2 b 4. Taking the second derivative, we get h (b) = 6x 2 b + 6 + 3x 2 ( 4)b 5 = 6x 2 b + 6 2x2 Thus, h () = 3x 2 + 6 + 3x 2 = 6x 2 + 6. Also, h (x) = 6x 3 + 6 2 x. 3. The graph of f (x) is shown below. At which of the marked points is (a). f(x) the least? (b). f(x) the greatest? (c). f (x) the least? (d). f (x) the greatest? (e). f (x) the least? (f). f (x) the greatest? (a), (b). Since f (x) > 0 over the interval of x-coordinates of interest (spanning all four points), we know that f(x) is increasing over the interval. Thus, the least value of f(x) occurs at A and the greatest occurs at D. (c), (d). Among the four points marked, f (x) is greatest at D and is least at C. (e), (f). Among the four points marked, the slope, f (x), of f (x) is greatest at C (where the slope is positive) and is least at B (where the slope is negative). 2. At a time t after it is thrown up in the air, a ball is at a height of f(t) = ct 2 +8t+3 meters, where c is a constant. We are told that at time t = the ball is accelerating downwards at 6 meters/sec 2. (a). What is the velocity of the ball at time t =? Is the ball going up or going down at time t =? The velocity at time t is given by f (t) = 2ct + 8. The acceleration at time t is given by f (t) = 2c (it is constant). Since we are told that at time t =, the ball is accelerating downwards at 6 meters/sec 2, we know that f () = 6, so c = 3. We are asked for f () = 2( 3)() + 8 = 2. Since f () > 0, we know that the ball is going up at t =. (b). How high does the ball go? The ball reaches it highest point when the velocity is 0, i.e., when f (t) = 0, or 6t + 8 = 0, which means that t = 4/3. The height at time t = 4/3 is f(4/3) = 3 (4/3) 2 + 8 (4/3) + 3 = 25/3 meters. 3. (a). Let f(x) = x2 e x 2x 3 +e x ln( x). Find f (x). For the first term, we apply the quotient rule, and for the second we apply the chain rule: f (x) = (2x3 + e x )(x 2 e x ( ) + 2xe x ) x 2 e x (6x 2 + e x ) (2x 3 + e x ) 2 x (/2)x /2 (b). Let f(x) = cos(3tan(3x)). Find f (x). b 5. 3
We aply the chain rule: f (x) = ( sin(3tan(3x))(3sec 2 (3x))3 4. Suppose xy y 2 = e x2. Compute dy dx. We differentiate implicitly, on both side of the equation, with respect to x: (xy + y) 2yy = e x2 2x, which implies that y = ex2 2x y x 2y 5. (a). Suppose that h(x) = 2x + g(x 2 ) and that g (u) = 3u 2. Find h (x). We take the derivative with respect to x, applying the chain rule to the g(x 2 ) term: h (x) = 2 + g (x 2 ) 2x = 2 + 3(x 2 ) 2 2x = 2 + 6x 5. (b). Let f(x) = 3e 2x+. Find the 00th derivative, f (00) (x). First we compute f (x) = 3e 2x+ 2 = 3 2 e 2x+. Then we compute f (x) = 3 2 2 e 2x+, and then f (x) = 3 2 3 e 2x+. Each time we take the derivative again, we pick up another factor of 2, from the chain rule (the derivative of 2x + ). Thus, f (00) (x) = 3 2 00 e 2x+. 6. The average cost per item, C, in dollars, of manufacturing a quantity q of cell phones is given by C = (a/q) + b, where a, b, are positive constants. (a). Find the rate of change of C as q increases. What are its units? (b). If production increases at a rate of 00 cell phones per week, how fast is the average cost changing? Is the average cost increasing or decreasing? (a). Since C(q) = (a/q)+b, we get C (q) = (a/q 2 ) dollars per cell phone is the rate of change of C as q increases. (b). Now, considering q to be a function of t, q(t), we can compute the derivative of C with respect to t: dc dt = a q 2 dq dt = a q 2 00, in units of dollars per week (since we are told that dq dt is 00 cell phones per week the production is increasing at the rate of 00 cell phones per week). Since dc/dt is negative (since a. q 2 are positive), we see that the average cost is decreasing. 7. (a). Compute t + 2te /t 20t. Rewriting the expression as a ratio, we get: t + 2te/t 20t = t + 2e /t 20, /t which has the form 8 0, so the it does not exist (one could say that the it is ). (a ). This is the version that was intended: Compute t + 2te /t 2t. Rewriting the expression as a ratio (whose it form is 0 0 as x ), and then applying l Hopital s rule, we get: t + 2te/t 2t = t + which has the form 0 0, so we apply l Hopital s rule to get: 2e /t 2, /t 2e /t 2 2e /t ( /t 2 ) = t + /t t + ( /t 2 = ) t + 2e/t = 2. (b). Compute t 0 t sin t. Rewriting the expression as a ratio (whose it form is 0 0 as x ), and then 4
applying l Hopital s rule, we get: t 0 t sin t = sint t cos t = t 0 t sin t t 0 t cos t + sint, which still has the form 0 0, so we apply l Hopital s rule again to get t 0 cos t t cos t + sin t = t 0 sin t t( sin t) + cos t + cos t = 0 2 = 0. 8. A rectangular beam is cut from a cylindrical log of radius 30 cm. The strength of a beam of width w and height h is proportional to wh 2. Find the width and height of the beam of maximum strength. Draw a picture! (This is a problem from the textbook, so you can see the picture there.) Then we see that (w/2) 2 + (h/2) 2 = 30 2, so h 2 = 4(900 w 2 /4) = 3600 w 2. Now, the strength, f(w), is given by f(w) = wh 2 = w(3600 w 2 ). We compute the derivative: f (w) = 3600 3w 2. Setting f (w) = 0, we get 3600 = 3w 2, so w = 200 and w = 200 are critical points. (Only w = 200 is physically meaningful.) We can do the second derivative test: f (w) = 6w, so f ( 200) < 0, and we see that w = 200 gives a local max, which is in fact a global max for w (0, ) (since f(0) = 0 is not greater, and w f(w) = is much less than f( 200)). When w = 200, h = 3600 ( 200) 2 = 2400 is the corresponding height. 9. Consider the function f(x) = 3x 5 5x 3. (a). Determine the critical points and classify each as a local max, local min, or neither. We compute f (x) = 5x 4 5x 2. Thus, the critical points are given by solving 5x 4 5x 2 = 0, which gives 5x 2 (x 2 ) = 0, so x = 0,, are the critical values of x. In order to classify the critical points, we can apply the first derivative test: (i). f (x) does not change sign at x = 0 (it is negative for x = 0.0 and also for x = 0.0), so x = 0 is not a local min or local max; (ii). f (x) changes from positive to negative at x = (f (.0) > 0, while f (.99) < 0), so x = is a local max; (iii). f (x) changes from negative to positive at x = (f (0.99) < 0, while f (.0) > 0), so x = is a local min. (Alternatively, we can apply the second derivative test, using f (x) = 60x 3 30x: (i). f (0) = 0, so we cannot tell (use first derivative test instead); (ii). f ( ) = 30 < 0, so x = is a local max; (iii). f () = 30 > 0, so x = is a local min.) (b). On which intervals is the function concave up? Concave down? Identify any inflection points. Setting f (x) = 60x 3 30x to 0, we get x = 0 or x = /2 or x = /2. Thus, we consider the corresponding intervals, /2), ( /2,0), (0, /2), and ( /2, ). We can check that f (x) is positive on the interval (, /2), where f is concave down. We can check that f (x) is negative on the interval ( /2,0), where f is concave up. At x = /2, there is an inflection point. We can check that f (x) is positive on the interval (0, /2), where f is concave down. At x = 0, there is an inflection point. We can check that f (x) is negative on the interval ( /2, ), where f is concave up. At x = /2, there is an inflection point. (c). Find the global max/min on the interval x [ 0,0]. The critical points (x = 0,,) lie in the interval [-0,0]; there is a local max at x =, where f( ) = 3 + 5 = 2, and there is a local min at x =, where f() = 3 5 = 2. We must also check the endpoints of the interval: f( 0) = 300,000+5000 = 295,000 and f(0) = 300,000 5000 = 295,000. Thus, the global max on the interval is at x = 0 (f(0) = 295,000), and the global min on the interval is at x = 0 (f( 0) = 295,000). 20. Joe runs a marathon. He gets more and more tired, so his speed decreases over time. His speed at certain times is given below. Time (min) 0 5 30 45 60 75 90 Speed (mph) 2 2 0 9 8 0 5
Give upper and lower bounds on the distance that Joe runs during the first hour. Let v(t) be the velocity at time t. Since v(t) is a decreasing function, we know that the left-hand-sum is an upper bound on total distance traveled, and the right-hand-sum is a lower bound on total distance traveled. We are interested in the first hour, t (0,), so the left-hand-sum is given by (2 + 2 + + 0) = 0.25, 4 and the right-hand-sum is given by (2 + + 0 + 9) = 0.5, 4 so we know that the distance d travelled in the first hour obeys 0.25 d 0.5 (Note that if we knew the actual function v(t), then the exact distance traveled in the first hour is given by 0 v(t)dt.) 6